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    4.5 Autoregressive Processes AR (P) : Definition 4.7. An

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    harry
    The document summarizes autoregressive (AR) time series models. It defines an AR(p) model as explaining the current value of a time series (Xt) as a linear function of its past p values plus white noise. It provides different notations for writing AR models, including vector and backshift operator notation. It then focuses on the special case of AR(1) models, showing how they can be written as an infinite combination of past white noise values, making AR(1) models stationary time series.

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    4.5 Autoregressive Processes AR (P) : Definition 4.7. An

    Uploaded by

    harry
    48 views2 pages
    The document summarizes autoregressive (AR) time series models. It defines an AR(p) model as explaining the current value of a time series (Xt) as a linear function of its past p values plus white noise. It provides different notations for writing AR models, including vector and backshift operator notation. It then focuses on the special case of AR(1) models, showing how they can be written as an infinite combination of past white noise values, making AR(1) models stationary time series.

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    ts

    Original Title

    TS_Chapter4_5-1-2

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    The document summarizes autoregressive (AR) time series models. It defines an AR(p) model as explaining the current value of a time series (Xt) as a linear function of its past p values plus white noise. It provides different notations for writing AR models, including vector and backshift operator notation. It then focuses on the special case of AR(1) models, showing how they can be written as an infinite combination of past white noise values, making AR(1) models stationary time series.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    48 views2 pages

    4.5 Autoregressive Processes AR (P) : Definition 4.7. An

    Uploaded by

    harry
    The document summarizes autoregressive (AR) time series models. It defines an AR(p) model as explaining the current value of a time series (Xt) as a linear function of its past p values plus white noise. It provides different notations for writing AR models, including vector and backshift operator notation. It then focuses on the special case of AR(1) models, showing how they can be written as an infinite combination of past white noise values, making AR(1) models stationary time series.

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    74 CHAPTER 4.

    STATIONARY TS MODELS

    4.5 Autoregressive Processes AR(p)


    The idea behind the autoregressive models is to explain the present value of the
    series, Xt , by a function of p past values, Xt−1 , Xt−2 , . . . , Xt−p .

    Definition 4.7. An autoregressive process of order p is written as

    Xt = φ1 Xt−1 + φ2 Xt−2 + . . . + φp Xt−p + Zt , (4.20)

    where {Zt } is white noise, i.e., {Zt } ∼ W N(0, σ 2 ), and Zt is uncorrelated with
    Xs for each s < t.

    Remark 4.12. We assume (for simplicity of notation) that the mean of Xt is zero.
    If the mean is E Xt = µ 6= 0, then we replace Xt by Xt − µ to obtain

    Xt − µ = φ1 (Xt−1 − µ) + φ2 (Xt−2 − µ) + . . . + φp (Xt−p − µ) + Zt ,

    what can be written as

    Xt = α + φ1 Xt−1 + φ2 Xt−2 + . . . + φp Xt−p + Zt ,

    where
    α = µ(1 − φ1 − . . . − φp ).

    Other ways of writing AR(p) model use:

    Vector notation: Denote


    φ = (φ1 , φ2 , . . . , φp )T ,
    Xt−1 = (Xt−1 , Xt−2 , . . . , Xt−p )T .

    Then the formula (4.20) can be written as

    Xt = φT Xt−1 + Zt .

    Backshift operator: Namely, writing the model (4.20) in the form

    Xt − φ1 Xt−1 − φ2 Xt−2 − . . . − φp Xt−p = Zt ,

    and applying BXt = Xt−1 we get

    (1 − φ1 B − φ2 B 2 − . . . − φp B p )Xt = Zt
    4.5. AUTOREGRESSIVE PROCESSES AR(P) 75

    or, using the concise notation we write

    φ(B)Xt = Zt , (4.21)

    where φ(B) denotes the autoregressive operator

    φ(B) = 1 − φ1 B − φ2 B 2 − . . . − φp B p .

    Then the AR(p) can be viewed as a solution to the equation (4.21), i.e.,
    1
    Xt = Zt . (4.22)
    φ(B)

    4.5.1 AR(1)
    According to Definition 4.7 the autoregressive process of order 1 is given by

    Xt = φXt−1 + Zt , (4.23)

    where Zt ∼ W N(0, σ 2 ) and φ is a constant.

    Is AR(1) a stationary TS?

    Corollary 4.1 says that an infinite combination of white noise variables is a sta-
    tionary process. Here, due to the recursive form of the TS we can write AR(1) in
    such a form. Namely
    Xt = φXt−1 + Zt
    = φ(φXt−2 + Zt−1 ) + Zt
    = φ2 Xt−2 + φZt−1 + Zt
    ..
    .
    k−1
    X
    = φk Xt−k + φj Zt−j .
    j=0

    This can be rewritten as


    k−1
    X
    φk Xt−k = Xt − φj Zt−j .
    j=0

    What would we obtain if we have continued the backwards operation, i.e., what
    happens when k → ∞?

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